## Historical and Philosophical Approach to Curriculum Theory

learning theories as supporting evidence and the curiculum should be based on different range of maths level and what inpacts it has on the learners for the functional and life skills.

Curriculum theory is primarily an academic discipline that is mainly concentrated on the examination and shaping of the educational curricula. The study of curriculum theory is very wide and it includes the historical analysis of the curriculum methods and the methods to view the contemporary educational policies and curriculum. Many scholars have presented different views on the curriculum among which the views of the scholars Herbert Kliebard and Michael Stephen Schiro are considered to be very significant.

On one hand when Kliebard’s view regarding curriculum theory has an inclination towards the historical approach Schiro’s theory has a more philosophical approach. The historical approach given by Kliebard examines the forces which are at work and that shaped the American curriculum. It was more prevalent during 1893 and 1958. Schiro’s theory examines the ideologies of curriculum that influenced the American curriculum. This practice was prevalent during the 1890-2007 (Pinar, 2004). In this theory Kliebard has discussed four curriculum groups which are called humanist, developmentalist, efficiency and social meliorists. On the other hand, the theories of Schiro are social efficiency ideology, learner-centered ideology, social reconstruction ideology and scholar academic ideology.

Proceeding to the curriculum theory of another scholar John Dewey it can be noticed that his theory is such that the students are able to deal efficiently with the contemporary world. Hence, he stated that curriculum should have unfinished abstractions however, they should include the preconceptions of the children and should also include the children’s views of their personal world. According to Dewey, there are four instincts that can describe the characteristics of any child’s behavior. These four instincts are social, constructive, expressive and artistic. The curriculum theory should be made in accordance to the logical sense of the child and the surroundings.

Dewey further stated that the curriculum should be such as to present the capacity level of the child. The primary task is then to find the level, to motivate the material, to allow the routine to make appropriate learning experience for ordinary activities that is not connected to any prior knowledge of the students. Dewey gives an elaborate definition of curriculum theory as a summary or a map or a view of experiences that has been arranged previously to serve as a guide for future experience and give direction. It also enables control, save efforts, prevent aimless wanderings and also point out the ways to lead more quickly and certainly to get a desired result.

Further according to Dewey the curriculum theory can be summarized as an organizational plan that can guide and also be adaptable. It takes into consideration the present knowledge level of the student and also figures out the appropriate path. These occupations were used by Dewey to connect with the small versions of the basic activities of classroom life.

Among the theories of curriculum the educational theory is one where apparently all the levels of education would be concerned in providing a liberating experience since the education would emphasize in promoting freedom of thought, independence, political and social empowerment, acceptance of opinions with no discrimination based on caste, creed, race, sex or nationality (Paraskeva, 2011). The total curriculum theory is significant and it is the rationale of this curriculum that is of priority. The total curriculum is one that includes the total programme of the educational institution. It is not restricted to the syllabus of the school. The next curriculum is the hidden curriculum that includes those things that the students learn at the schools depending on the way in which the school is organized and planned and also by the materials provided (Pratte, 1981). Sometimes these things are not included explicitly in the planning and are generally communicated to the pupils in an accidental manner.

## Different Views on Curriculum by Kliebard and Schiro

The planned and received curriculum can be divided to mean the ones that lay down the prospectus and syllabuses for planned curriculum and the one that is the reality of the students’ experience is the received curriculum (Schiro, 2008). And finally the formal and the informal curriculum which is primarily a distinction between a formal activity which the time table of the school states for specific studies and informal activity that includes sports, excursions, societies or clubs which are together generally termed as ‘extra-curricular’ activities.

Source: (infed.org, 2013)

The above diagram depicts the Aristotle’s knowledge of three disciplines based on the study of curriculum. Here the body of knowledge is transmitted as ‘canon’, the process and praxis are the practical part and the technical part is the product model.

In order to analyse a particular aspect of curriculum, the researcher has chosen the different levels of math to study this aspect of curriculum.

In the present times, the education of mathematics may be considered as the practice of teaching and learning different aspects of mathematics and also the scholarly research. The levels of mathematics are different at different times, depending on the different cultures, and different countries with the aim to achieve a number of objectives (Davis, 1967).

These objectives are as follows:

- Teaching and learning the basic skills of numerical for the students;
- Teaching practical mathematics including arithmetic, elementary algebra, plane and solid geometry and trigonometry to the students;
- Teaching mathematical concepts such as set and function at an early age;
- Teaching selected areas of mathematics such as axiomatic system or deductive reasoning; or calculus as an instance of intellectual achievements;
- Teaching advanced mathematics to those who wish to have a career in Science, Engineering, Mathematics and Technology fields;
- Teaching heuristics and problem solving strategies.

These different levels of mathematics are taught at different ages and different sequences in various countries. Sometimes class is taught at an early age and typical in any honors class.

Elementary mathematics is taught is most countries in the same manner with minor differences. For instance in the United States fractions are taught from 1^{st} grade and in other countries it is taught much later. This is primarily because the metric system in their countries does not require the young to be familiar with such things (CHOPPIN, 2009). Generally the countries require very few topics to be dealt in depth unlike United States. These include topics such as addition, subtraction, multiplication, and division and also pre-algebra (Davis, 1967).

In some countries the math’s subjects such as geometry, analysis and algebra are taught in separate courses in separate classes in the high school. This subject is integrated in most countries as all branches of the subjects are taught in each year to the students (CHOPPIN, 2009). Sometimes the students are given an option to choose between courses of mathematics. Further students who take up science as their curricula they need to study calculus and trigonometry that includes integral calculus, complex numbers, exponential functions and logarithms and analytical geometry. Also generally probability and statistics was also taught in the secondary classes along with infinite series.

Students of science and engineering in various universities and colleges need to take up multi-variable calculus, linear algebra and differential calculus. In the study of majors applied mathematics is studied and in computer science mathematics includes graph theory, probability, proofs and permutations (Schiro, 2008).

Innovative programmes in mathematics will necessarily change the contemporary practices and thoughts on the present perception of knowledge and the professional strategies of teachers. This learning and teaching need to occur in a proper social context. Hence construction of knowledge of children must be in an appropriate integrated and holistic fashion (Mathematics learners and mathematics textbooks: a question of identity? Whose curriculum? Whose mathematics?, 2010). So the social context in which the child learns and the teacher teaches becomes extremely significant. The implementation of the mathematics curricula should be in a local context since this implementation is dependent on the high adoption rates by the schools and results of examinations. The economic and the political aspects of the mathematics curriculum must also be considered.

## John Dewey's Approach to Curriculum Theory

It is interesting to know that mathematical relationships can be used responsibly to address certain human rights issues such as the economic al and political relationships. Learners when they develop mathematical sense to represent and misrepresent trends, manipulate data, critically analyse data, make predictions and interpret chance variations. Hence this allows the learners to not only learn mathematics but also utilize these learning in meaningful social, political and economic activities (Mathematics learners and mathematics textbooks: a question of identity? Whose curriculum? Whose mathematics?, 2010). These curriculums happen to be a critical part of mathematics curriculum. This acts as a challenge in the mainstream mathematical education and likewise new and innovative curricula are developed in order to give mathematics a new dimension based on social, political and economic aspect.

In the present times, curriculum evaluation has become increasingly popular generating a lot of interest. There are a number of evaluation models which exist in recent times such as the Bradley’s Effectiveness Model, Scriven’s Goal-free Model, Stake’s Responsive Model and Eisner’s Connoisseurship Model.

The Effectiveness Model is based on the vertical and horizontal curriculum, broad involvement, long-range planning, decision making clarity etc. The objective centered Model is an early curriculum evaluation model which focuses on strength and weaknesses of curriculum. However, the problem is that this model focuses on the assessment without the suggestions.

The next model which is the Context, Input, Process, Product Model is more relevant for the educational leaders as this model gives assessment data and makes evaluation much easier for decision making. In this model there are a number of steps that needs to be followed. These include identification of the decisions, identification of the data, collection of the data and determination of the quality.

The goal free model aims to divert the attention of the administrators and evaluators to the significance of the unintended effects of the evaluation program. The next model which is the responsive model stresses on the responsibilities of the stakeholders on whom the evaluation effects. The last model is the connoisseurship model which is based o criticism and appreciation. According to this model the evaluation process should be based on these two vital aspects of criticism and appreciation.

Other than these models there are more models present for curriculum evaluation. However, there are very few that could be effectively applied to the development projects of mathematics curriculum. Mathematical projects can be evaluated based on the structuralist project. This approach is based on the analysis, creative and systematic application of social interactions and structures in classrooms.

Mathematics curriculum can also be based on the new-math approach. This approach of evaluation is based on content evaluation. This particular method of teaching depended on the factual information and performing on the routine calculations. This approach of evaluation was not very effective for mathematics curriculum as there were deficiencies seen in the standardized test performances.

Another evaluation procedure on which the mathematics curriculum can be based is the behaviorist approach. Eisenberg had argued that this approach of evaluation since it equates education with evaluation generally miss the spirit of the actual discipline of mathematics. Even this approach of evaluation has been widely criticized by the scholars. Yet the objective model of this behavioral approach gained significance and has also been used in mathematical assessment of curriculum. In this particular approach of evaluation study it was observed that the degree of individualism was high. Observation and interview studies made children more aware than evaluations based on paper and pencil.

## Different Levels of Mathematics Taught Across Various Countries

Another evaluation procedure was the formative approach. This particular approach based on mathematical curriculum focuses on the fact that school education should aim at incorporating personality traits in students. These personality traits include cognitive and motivational activities, affective attitudes or intelligence and performance motivation. Hence the curriculum developer needs to search and match properly the contents which develop these personality traits. The projects which are based on this formative approach would be more illustrative and suggestive instead of being more definitive and comprehensive.

Finally the last approach for mathematics curriculum would be the integrated-teaching approach. This approach analyses the particular problem in hand and also tries to erase the boundaries existing between mathematics and other disciplines. This can become a challenge for teachers if they are trained properly in this discipline. This can be considered as the most successful evaluation procedure since it was observed that students who were taught under this procedure showed very good growth than those students who were not taught under this approach.

The modern day mathematics education is based on the investigation on numbers, data and space, connected mathematics etc. In the present days children are explicitly being taught formulas and algorithms which more of an inquiry based approach. The advocates of the modern mathematics state that children should focus on the development of conceptual understanding of the problems rather than just drill on formulas and algorithms. If the understanding is clear then students would develop fluency in the understanding of the concept and calculations. However, this present day approach of mathematics curriculum is facing a lot of criticism from different corners. Scholars state that students must primarily develop skills on computation before they are made to understand the deep concepts of mathematics. Proper practice and memorization of these skills are necessary before they become automatic for the students. When time is spent in practicing skills it is more worthwhile rather than spending time in investigating on mathematical concepts. In order to estimate answers it is essential to have very strong foundational skills. To understand the concepts of mathematics it is essential to have a solid base of mathematical knowledge of the various tools of subject.

In the world of academia the traditional methods of mathematics is still used widely. Further, modern mathematics excessively depends on the new technologies such as calculators. Use of such technologies needs to be restricted to some extent till a particular age. Constructive methods when used for children make it unfamiliar with the adults hence making it difficult to work on or help with home work of the children. Students are required to learn the skills of flexible thinking in order to develop the practice to solve problems. Critics state these methods of flexible thinking can be developed only after they have mastered the foundational skills of mathematics.

Recommendation and Conclusion

As concluding remarks it can be stated that this research study based on the curriculum theory shows the different aspects of the theories and also gives an elaborate analysis of a particular curriculum based on the level of mathematics. Throughout the study it can be observed due to the contemporary changes in the mathematics curriculum the study has become vastly different from the traditional approach. However, even now the involvement of mathematics and education researchers is limited in most cases and also is the examination on curriculum. Teachers should engage more in lesson study and that should be utilized to have a feedback mechanism and that would focus on the school mathematics curriculum.

References

CHOPPIN, J. (2009). Curriculum-Context Knowledge: Teacher Learning From Successive Enactments of a Standards-Based Mathematics Curriculum. *Curriculum Inquiry*, 39(2), pp.287-320.

Davis, R. (1967). *The changing curriculum: mathematics*. Washington: Association for Supervision and Curriculum Development, NEA.

infed.org, (2013). *Curriculum theory and practice*. [online] Available at: https://infed.org/mobi/curriculum-theory-and-practice/ [Accessed 22 Jan. 2015].

Mathematics learners and mathematics textbooks: a question of identity? Whose curriculum? Whose mathematics?. (2010). *Curriculum Journal*, 21(2), pp.235-235.

Paraskeva, J. (2011). *Conflicts in curriculum theory*. New York: Palgrave Macmillan.

Pinar, W. (2004). *What is curriculum theory?*. Mahwah, N.J.: Lawrence Erlbaum.

Pratte, R. (1981). Metaphorical Models and Curriculum Theory. *Curriculum Inquiry*, 11(4), p.307.

Schiro, M. (2008). *Curriculum theory*. Los Angeles, Calif.: Sage Publications.

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